Any idea what to add to oak stain to make it thick llike paint?

I can only think of a couple of ways you're going to get a decent match. I would try this on some wood scraps first. I would try painting it a light brown and blend in some other tones and then seal it with the same level of gloss as the rest of it. The other method I would try is using a darker stain to help the spindle blend in. Stains can be mixed. Oak darkens with age so you aren't going to get a good match with an oak stain. You might have to play with it a bit to get it to match. Small containers of stain are fairly expensive, so go with a very light, warm color like a light maple to mix with the oak. I mix tiny amounts at a time in a small glass jar or plastic dish, trying to get the tone I want. You may have to apply it more than once to get what you need. You can use steel wool between coats after the coat just applied is dry. That way the wood will take more stain. I've used both methods for small repairs I wanted to blend in.

Stain the spindle darker and sand

You don't describe "funny" outcome.... is it too light? Try rubbing with a darker stain and rub off pretty immediately to fill the wood pores, then go over it with the lighter stain. Make sure your stirring the stain well cause all the pigment will be on the bottom. It might also not match cause the top coat has yellowed over time giving an amberish glow to it. Add a little gold pigment to the top coat.

I know you can make chalk paint from grout powder and paint. Maybe you could add grout powder??

Spray the stain on, I have found that to be the only way to get the thickness of color.